Is there any difference between these two usages? I thought they both meant to walk, but someone told me (though not sure if they are right or not) that spazieren uses different grammar. I couldn’t find anything on it, so I thought I would post here to clear it up.
I don't see a grammatical difference, but a (slight) one in meaning. You could compare it to the difference between "walking" (maybe "strolling") and "taking a walk".
Er spaziert über den Platz.
He walks/strolls across the square.
Er geht im Park spazieren.
He takes a walk in the park.
Spazierengehen is more closed than spazieren.
One the one hand this means, one can say:
Ich gehe spazieren
and the sentence is complete (expressing i.e. your desire to talk a walk), whereas ich spaziere is a fragment only matching to a very primitive question like Was tust du gerade? (What are you doing there?)
One the other hand, Hennings example is the only type of extension possible at all. You can add a location or a time where the Spaziergang takes place, but that's it.
Sie spazierte zum Teich, kaufte sich ein Eis und schaute den Schwänen zu.
(She took a walk to the pond, bought a cone of ice cream and looked to the swans) in contrast specifies, where the walk was leading, a construction impossible to be squeezed into a sentence with spazierengehen.